(2x-3)(x+1)-(x-2)<0 eching

$$(2x-3)(x+1)-(x-2) < 0$$

x uchun yechish

$$x\in \left(\frac{1-\sqrt{3}}{2},\frac{\sqrt{3}+1}{2}\right)$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x-3 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-x-3-\left(x-2\right)<0$$

x-2 teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}-x-3-x+2<0$$

-2x ni olish uchun -x va -x ni birlashtirish.

$$2x^{2}-2x-3+2<0$$

-1 olish uchun -3 va 2'ni qo'shing.

$$2x^{2}-2x-1<0$$

Tengsizlikni yechish uchun chap tomon faktorini hisoblang. Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

$$2x^{2}-2x-1=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 2 ni, b uchun -2 ni va c uchun -1 ni ayiring.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

Hisoblarni amalga oshiring.

$$x=\frac{2±2\sqrt{3}}{4}$$

x=\frac{2±2\sqrt{3}}{4} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=\frac{\sqrt{3}+1}{2} x=\frac{1-\sqrt{3}}{2}$$

Yechimlardan foydalanib tengsizlikni qaytadan yozing.

$$2\left(x-\frac{\sqrt{3}+1}{2}\right)\left(x-\frac{1-\sqrt{3}}{2}\right)<0$$

Koʻpaytma manfiy boʻlishi uchun x-\frac{\sqrt{3}+1}{2} va x-\frac{1-\sqrt{3}}{2} qarama-qarshi belgilar boʻlishi kerak. x-\frac{\sqrt{3}+1}{2} musbat, x-\frac{1-\sqrt{3}}{2} manfiy boʻlganda, yechimni toping.

$$x-\frac{\sqrt{3}+1}{2}>0 x-\frac{1-\sqrt{3}}{2}<0$$

Bu har qanday x uchun xato.

$$x\in \emptyset$$

x-\frac{1-\sqrt{3}}{2} musbat, x-\frac{\sqrt{3}+1}{2} manfiy boʻlganda, yechimni toping.

$$x-\frac{1-\sqrt{3}}{2}>0 x-\frac{\sqrt{3}+1}{2}<0$$

Ikkala tengsizlikning mos yechimi – x\in \left(\frac{1-\sqrt{3}}{2},\frac{\sqrt{3}+1}{2}\right).

$$x\in \left(\frac{1-\sqrt{3}}{2},\frac{\sqrt{3}+1}{2}\right)$$

Oxirgi yechim olingan yechimlarning birlashmasidir.

$$x\in \left(\frac{1-\sqrt{3}}{2},\frac{\sqrt{3}+1}{2}\right)$$